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Properties

Label 2-2448-612.115-c0-0-1
Degree $2$
Conductor $2448$
Sign $0.978 - 0.208i$
Analytic cond. $1.22171$
Root an. cond. $1.10531$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.965 + 0.258i)3-s + (0.866 + 0.499i)9-s + (0.965 − 0.258i)11-s + (−0.5 − 0.866i)13-s + i·17-s − 1.41·19-s + (0.258 − 0.965i)23-s + (0.866 + 0.5i)25-s + (0.707 + 0.707i)27-s + (1.36 − 0.366i)29-s + (0.965 + 0.258i)31-s + 33-s + (−0.258 − 0.965i)39-s + (−0.707 + 1.22i)43-s + (−1.22 − 0.707i)47-s + ⋯
L(s)  = 1  + (0.965 + 0.258i)3-s + (0.866 + 0.499i)9-s + (0.965 − 0.258i)11-s + (−0.5 − 0.866i)13-s + i·17-s − 1.41·19-s + (0.258 − 0.965i)23-s + (0.866 + 0.5i)25-s + (0.707 + 0.707i)27-s + (1.36 − 0.366i)29-s + (0.965 + 0.258i)31-s + 33-s + (−0.258 − 0.965i)39-s + (−0.707 + 1.22i)43-s + (−1.22 − 0.707i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 - 0.208i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 - 0.208i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2448\)    =    \(2^{4} \cdot 3^{2} \cdot 17\)
Sign: $0.978 - 0.208i$
Analytic conductor: \(1.22171\)
Root analytic conductor: \(1.10531\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2448} (1951, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2448,\ (\ :0),\ 0.978 - 0.208i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.765917910\)
\(L(\frac12)\) \(\approx\) \(1.765917910\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.965 - 0.258i)T \)
17 \( 1 - iT \)
good5 \( 1 + (-0.866 - 0.5i)T^{2} \)
7 \( 1 + (0.866 - 0.5i)T^{2} \)
11 \( 1 + (-0.965 + 0.258i)T + (0.866 - 0.5i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + 1.41T + T^{2} \)
23 \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \)
29 \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \)
31 \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + (0.866 + 0.5i)T^{2} \)
43 \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \)
53 \( 1 + iT - T^{2} \)
59 \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \)
83 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 - T + T^{2} \)
97 \( 1 + (0.866 - 0.5i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.942193301660410812194896862420, −8.409463430970386789144910627322, −7.963359815905048743705080372340, −6.69860546412657486337511577002, −6.36489926499773911149438968514, −4.92679661523468461175976122923, −4.34059287753340781018046684315, −3.35347775250659260676086507980, −2.59726102866786692298854361157, −1.40234315621134033404718317464, 1.37551739855372979944317146701, 2.38400701475119244045066190826, 3.26736064880618035747038329469, 4.33545168543368200058577438929, 4.83267055825219169908105905739, 6.46302073244650297058209792313, 6.72346389806842472356693372240, 7.57859221760850254436684424581, 8.488299362139684661759549251910, 9.013622698105209108566165814084

Graph of the $Z$-function along the critical line